| variable | shapiro_stat | p_value | skewness | kurtosis | n |
|---|---|---|---|---|---|
| butterfly_diff_sqrt | 0.9893 | 0.6368 | 0.2251 | -0.3143 | 96 |
| butterfly_diff_95th_sqrt | 0.9880 | 0.5411 | 0.0120 | -0.6497 | 96 |
| butterfly_diff_top3_sqrt | 0.9875 | 0.5027 | 0.0266 | -0.6207 | 96 |
| butterfly_diff_95th | 0.9182 | 0.0000 | -0.3705 | 2.0466 | 96 |
| butterfly_diff_top3 | 0.9126 | 0.0000 | -0.0126 | 2.4706 | 96 |
| butterfly_diff | 0.8835 | 0.0000 | 0.3794 | 4.3938 | 96 |
| butterfly_diff_95th_sq | 0.6591 | 0.0000 | -1.2037 | 9.1440 | 96 |
| butterfly_diff_top3_sq | 0.6327 | 0.0000 | 0.6220 | 10.0799 | 96 |
| butterfly_diff_sq | 0.5307 | 0.0000 | 1.9996 | 19.7056 | 96 |
Dynamic Window GAM Analysis: Comprehensive Synthesis
1 Background
This document synthesizes results from two parallel GAM analyses investigating the relationship between weather conditions and monarch butterfly roost abandonment. Both analyses address feedback from Francis regarding temporal alignment of weather predictors with butterfly responses.
The original daily-level analysis used fixed 6am-6pm weather windows, which Francis identified as having a temporal logic issue:
“All of the metrics for wind, temperature and light, would need to be re-calculated for 24 hour periods that begin at the time of the highest count.”
Francis’s key points:
- Temporal alignment: Butterflies can only respond to weather after it occurs
- Biological timing: If max count occurred at 2pm on day t-1, the relevant weather window should start at 2pm, not 6am
- Roosting decisions: Weather from max count through sunset determines whether butterflies abandon the roost
1.1 Two-Window Comparison Approach
To address this feedback while exploring biological relevance, we conducted two parallel analyses:
Sunset Window Analysis (variable-length windows):
- Weather window:
time_of_max_count_t_1→last_observation_time_t(functional sunset) - Mean duration: 29.6 hours (range: 22.5-34.9 hours)
- Captures overnight conditions through sunset when roosting decisions finalize
- Tests: “Do conditions from peak count to sunset predict roost abandonment?”
- Data:
monarch_daily_lag_analysis_sunset_window.csv(n=103 lag pairs)
24-Hour Window Analysis (fixed-length windows):
- Weather window:
time_of_max_count_t_1→+24 hours - Fixed duration: exactly 24 hours for all observations
- Standardized exposure period following peak count
- Tests: “Do conditions in 24 hours following peak count predict roost abandonment?”
- Data:
monarch_daily_lag_analysis_24hr_window.csv(n=94 lag pairs)
Both analyses use identical modeling approaches to enable direct comparison of results across window definitions.
2 Response Variable Selection
We evaluated three butterfly difference metrics (max, 95th percentile, and top 3 mean) with three transformations each (untransformed, square root, and square) to determine which best approximates normality for GAM modeling.
2.1 Normality Testing Results
2.2 Selected Response Variable
butterfly_diff_sqrt was selected as the response variable for both analyses based on having the highest Shapiro-Wilk W statistic, indicating best approximation to normality.
This represents the square root transformation of the change in maximum butterfly count from day t-1 to day t. The transformation:
3 Candidate Predictor Variables
A total of 19 candidate predictors (3 baseline butterfly metrics + 16 weather/window metrics) were calculated within each observation’s dynamic window:
| Variable | Category | Description |
|---|---|---|
| max_butterflies_t_1 | Baseline | Maximum butterfly count on previous day |
| butterflies_95th_percentile_t_1 | Baseline | 95th percentile count on previous day |
| butterflies_top3_mean_t_1 | Baseline | Mean of top 3 counts on previous day |
| temp_min | Temperature | Minimum temperature in lag window |
| temp_max | Temperature | Maximum temperature in lag window |
| temp_mean | Temperature | Mean temperature in lag window |
| temp_at_max_count_t_1 | Temperature | Temperature at time of maximum count on t-1 |
| hours_above_15C | Temperature | Cumulative Hours ≥15°C in lag window |
| degree_hours_above_15C | Temperature | Cumulative degree-hours above 15°C |
| wind_avg_sustained | Wind | Mean sustained wind speed in lag window |
| wind_max_gust | Wind | Maximum wind gust in lag window |
| wind_gust_sum | Wind | Sum of all gust measurements in lag window |
| wind_gust_sum_above_2ms | Wind | Sum of gusts >2 m/s (threshold-based wind exposure) |
| wind_gust_hours | Wind | Gust-hours (integral approximation of wind exposure over time) |
| wind_minutes_above_2ms | Wind | Minutes with wind speed ≥2 m/s |
| wind_gust_sd | Wind | Standard deviation of gust speeds |
| wind_mode_gust | Wind | Most frequent gust speed (modal wind conditions) |
| sum_butterflies_direct_sun | Sun Exposure | Sum of butterflies observed in direct sunlight across entire lag window |
| lag_duration_hours | Window | Duration of lag window in hours. Used only for sunset analysis as a control variable. |
3.1 Full Correlation Matrix (All Candidates)
The full correlation matrix reveals substantial multicollinearity among candidate predictors, particularly within variable categories:
These patterns guided predictor selection to avoid multicollinearity issues.
3.2 Selected Predictors Correlation Matrix
Based on correlation structure, biological relevance, and variance inflation considerations, we selected 5 final predictors representing distinct aspects of weather exposure:
Final predictor set:
temp_min- Minimum temperaturetemp_max- Maximum temperaturetemp_at_max_count_t_1- Temperature at time of peak countwind_max_gust- Maximum wind gust, matching 30 minute analysissum_butterflies_direct_sun- Total minutes butterflies exposed to direct sun
3.2.1 Wind Gust Mode: An Excluded Variable
wind_gust_mode was initially considered as a candidate predictor due to its low correlation with other wind metrics (r < 0.3 with wind_max_gust). However, wind_gust_mode had to be excluded from model fitting due to having only 5 unique values across all observations. This extreme discretization caused model convergence issues.
4 Model Building Strategy
Both analyses employed an identical systematic model-building approach to ensure comparability.
4.1 Model Space Exploration
We fit 76 models spanning a hierarchy of complexity:
- Null models (2): Intercept only, with smooth vs. linear baseline
- Single predictor models (10): Each of 5 predictors, smooth vs. linear
- Interaction-only models (20): All 2-way interactions, smooth vs. linear
- Additive models (14): Multiple predictors without interactions, smooth vs. linear
- Main + Interaction models (20): Additive + selected interactions, smooth vs. linear
- Complex models (8): All temperatures, all temps + wind, etc.
- Full models (2): All predictors + all interactions, smooth vs. linear
4.2 GAM Structure
All models were fitted using gamm() (Generalized Additive Mixed Model) with the same core structure:
gamm(
butterfly_diff_sqrt ~ [baseline controls] + [weather predictors],
data = model_data,
random = list(deployment_id = ~1),
correlation = corAR1(form = ~ observation_order_t | deployment_id),
method = "REML"
)Key components:
- Response:
butterfly_diff_sqrt(square root transformed change in maximum count) - Baseline controls:
max_butterflies_t_1(both analyses) +lag_duration_hours(for sunset analysis only) - Random effects:
random = list(deployment_id = ~1)accounts for repeated measures - Temporal autocorrelation: AR(1) structure within deployments
- Weather predictors: Smooth
s()or tensor interactionti()terms - Family: Gaussian with identity link (justified by normality of transformed response)
4.3 Overfitting Controls & Model Selection
Earlier analyses selected full models despite df/n ≈ 0.5. To prevent overfitting with small sample sizes (n=94-103):
- Primary criterion: AICc (corrected AIC for small samples)
- Overfitting flags: df/n > 0.3 (High risk), df/n > 0.2 (Moderate risk)
- Cross-validation: Leave-one-deployment-out CV on top 10 models
- Alternative criteria: BIC, AIC weights, ΔAICc
Selection: Best model = lowest AICc; competitive models ΔAICc < 2
5 Sunset Window Analysis Results
5.1 Model Comparison
The table below shows the top 5 models ranked by AICc for the sunset window analysis:
| model | description | AICc | delta_AICc | weight_AICc | df | obs_per_param | overfitting_risk |
|---|---|---|---|---|---|---|---|
| M31 | Interaction: wind_max_gust × sum_butterflies_direct_sun (smooth) | 640.22 | 0.00 | 0.7993 | 11 | 8.73 | Low |
| M51 | Additive + Interaction: wind_max_gust + sum_butterflies_direct_sun + wind_max_gust×sum_butterflies_direct_sun (smooth) | 644.13 | 3.90 | 0.1135 | 15 | 6.40 | Low |
| M45 | Additive + Interaction: temp_max + sum_butterflies_direct_sun + temp_max×sum_butterflies_direct_sun (smooth) | 648.52 | 8.29 | 0.0126 | 15 | 6.40 | Low |
| M32 | Interaction: wind_max_gust × sum_butterflies_direct_sun (linear) | 648.85 | 8.63 | 0.0107 | 9 | 10.67 | Low |
| M19 | Interaction: temp_min × sum_butterflies_direct_sun (smooth) | 649.34 | 9.12 | 0.0084 | 11 | 8.73 | Low |
5.2 Best Model Summary
Model M31: wind_max_gust × sum_butterflies_direct_sun (smooth interaction only)
5.3 Model Summary
Family: gaussian
Link function: identity
Formula:
butterfly_diff_sqrt ~ max_butterflies_t_1 + lag_duration_hours +
ti(wind_max_gust, sum_butterflies_direct_sun)
Parametric coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 9.426342 5.516001 1.709 0.0911 .
max_butterflies_t_1 -0.029745 0.005963 -4.988 3.12e-06 ***
lag_duration_hours -0.190296 0.178387 -1.067 0.2891
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Approximate significance of smooth terms:
edf Ref.df F p-value
ti(wind_max_gust,sum_butterflies_direct_sun) 6.679 6.679 4.097 0.000832 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
R-sq.(adj) = 0.397
Scale est. = 41.056 n = 965.4 Partial Effects Plots
5.5 Model Diagnostics
5.6 Autocorrelation Diagnostics
The ACF and PACF plots show residual autocorrelation structure after accounting for AR(1) correlation structure in the model.
6 24-Hour Window Analysis Results
6.1 Model Comparison
The table below shows the top 5 models ranked by AICc for the 24-hour window analysis:
| model | description | AICc | delta_AICc | weight_AICc | df | obs_per_param | overfitting_risk |
|---|---|---|---|---|---|---|---|
| M31 | Interaction: wind_max_gust × sum_butterflies_direct_sun (smooth) | 636.33 | 0.00 | 0.5073 | 9 | 10.44 | Low |
| M23 | Interaction: temp_max × wind_max_gust (smooth) | 639.92 | 3.59 | 0.0844 | 9 | 10.44 | Low |
| M29 | Interaction: temp_at_max_count_t_1 × sum_butterflies_direct_sun (smooth) | 640.76 | 4.42 | 0.0555 | 9 | 10.44 | Low |
| M19 | Interaction: temp_min × sum_butterflies_direct_sun (smooth) | 641.11 | 4.78 | 0.0466 | 9 | 10.44 | Low |
| M51 | Additive + Interaction: wind_max_gust + sum_butterflies_direct_sun + wind_max_gust×sum_butterflies_direct_sun (smooth) | 641.72 | 5.39 | 0.0342 | 13 | 7.23 | Low |
6.2 Best Model Summary
Model M31: wind_max_gust × sum_butterflies_direct_sun (smooth interaction only)
6.3 Model Summary
Family: gaussian
Link function: identity
Formula:
butterfly_diff_sqrt ~ s(max_butterflies_t_1, k = 5) +
ti(wind_max_gust, sum_butterflies_direct_sun)
Parametric coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.099 1.101 -0.998 0.321
Approximate significance of smooth terms:
edf Ref.df F p-value
s(max_butterflies_t_1) 2.125 2.125 13.757 1e-05 ***
ti(wind_max_gust,sum_butterflies_direct_sun) 3.537 3.537 3.594 0.0196 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
R-sq.(adj) = 0.232
Scale est. = 52.881 n = 94Note: The 24-hour window has constant duration (24 hours for all observations), so lag_duration_hours is not included as a predictor.
6.4 Partial Effects Plots
6.5 Model Diagnostics
6.6 Autocorrelation Diagnostics
The ACF and PACF plots show residual autocorrelation structure after accounting for AR(1) correlation structure in the model.
7 Window Comparison: Convergent Results
Both the sunset window (variable-length, ending at functional sunset) and 24-hour window (fixed-length, standardized duration) analyses yielded the same best model:
M31: wind_max_gust × sum_butterflies_direct_sun (smooth interaction)
This convergence provides strong evidence that the wind × sun interaction is robust across different temporal window definitions.
8 Model Fitting Notes
8.1 Convergence
All 76 models (in both analyses) converged successfully with the following exceptions:
- Models including
wind_gust_modefailed to converge (variable excluded) - Full models (M71, M72) converged but flagged as high overfitting risk (df/n > 0.4)
8.2 Computational Details
- Software: R 4.x, mgcv package for GAM fitting
- Optimization: Outer iteration with GCV/REML for smoothness selection
- Basis dimensions: Default k values, checked via
gam.check() - AR1 correlation: Fitted via
gamm()using nlme correlation structures
8.3 Data Filtering
Both analyses filtered observations to:
metrics_complete >= 0.95(≥95% weather data completeness)- Complete cases for all 5 selected predictors
- Sunset window: n = 103 → 96 after filtering
- 24-hour window: n = 94 → 94 (no additional filtering)
9 Appendix: Full Model Comparison Tables
9.1 Sunset Window: All 76 Models
| model | description | AICc | delta_AICc | weight_AICc | baseline_type | model_category | overfitting_risk |
|---|---|---|---|---|---|---|---|
| M31 | Interaction: wind_max_gust × sum_butterflies_direct_sun (smooth) | 640.22 | 0.00 | 7.99e-01 | Smooth | Interaction only | Low |
| M51 | Additive + Interaction: wind_max_gust + sum_butterflies_direct_sun + wind_max_gust×sum_butterflies_direct_sun (smooth) | 644.13 | 3.90 | 1.13e-01 | Smooth | Main + Interaction | Low |
| M45 | Additive + Interaction: temp_max + sum_butterflies_direct_sun + temp_max×sum_butterflies_direct_sun (smooth) | 648.52 | 8.29 | 1.26e-02 | Smooth | Main + Interaction | Low |
| M32 | Interaction: wind_max_gust × sum_butterflies_direct_sun (linear) | 648.85 | 8.63 | 1.07e-02 | Linear | Interaction only | Low |
| M19 | Interaction: temp_min × sum_butterflies_direct_sun (smooth) | 649.34 | 9.12 | 8.36e-03 | Smooth | Interaction only | Low |
| M5 | Single: temp_max (smooth) | 649.46 | 9.23 | 7.90e-03 | Smooth | Single predictor | Low |
| M29 | Interaction: temp_at_max_count_t_1 × sum_butterflies_direct_sun (smooth) | 649.74 | 9.52 | 6.86e-03 | Smooth | Interaction only | Low |
| M43 | Additive + Interaction: temp_max + wind_max_gust + temp_max×wind_max_gust (smooth) | 650.12 | 9.89 | 5.68e-03 | Smooth | Main + Interaction | Low |
| M23 | Interaction: temp_max × wind_max_gust (smooth) | 650.17 | 9.95 | 5.53e-03 | Smooth | Interaction only | Low |
| M33 | Additive + Interaction: temp_min + temp_max + temp_min×temp_max (smooth) | 650.52 | 10.30 | 4.64e-03 | Smooth | Main + Interaction | Low |
| M39 | Additive + Interaction: temp_min + sum_butterflies_direct_sun + temp_min×sum_butterflies_direct_sun (smooth) | 651.39 | 11.17 | 3.00e-03 | Smooth | Main + Interaction | Low |
| M25 | Interaction: temp_max × sum_butterflies_direct_sun (smooth) | 651.78 | 11.56 | 2.47e-03 | Smooth | Interaction only | Low |
| M59 | Additive: temp_max + wind_max_gust (smooth) | 652.14 | 11.92 | 2.06e-03 | Smooth | Additive | Low |
| M55 | Additive: temp_min + temp_max (smooth) | 652.25 | 12.03 | 1.96e-03 | Smooth | Additive | Low |
| M15 | Interaction: temp_min × temp_at_max_count_t_1 (smooth) | 652.64 | 12.42 | 1.61e-03 | Smooth | Interaction only | Low |
| M17 | Interaction: temp_min × wind_max_gust (smooth) | 652.73 | 12.51 | 1.54e-03 | Smooth | Interaction only | Low |
| M11 | Single: sum_butterflies_direct_sun (smooth) | 652.82 | 12.60 | 1.47e-03 | Smooth | Single predictor | Low |
| M13 | Interaction: temp_min × temp_max (smooth) | 652.93 | 12.71 | 1.39e-03 | Smooth | Interaction only | Low |
| M1 | Null (smooth baseline) | 653.17 | 12.95 | 1.23e-03 | Smooth | Null | Low |
| M21 | Interaction: temp_max × temp_at_max_count_t_1 (smooth) | 653.48 | 13.26 | 1.06e-03 | Smooth | Interaction only | Low |
| M27 | Interaction: temp_at_max_count_t_1 × wind_max_gust (smooth) | 653.70 | 13.47 | 9.48e-04 | Smooth | Interaction only | Low |
| M49 | Additive + Interaction: temp_at_max_count_t_1 + sum_butterflies_direct_sun + temp_at_max_count_t_1×sum_butterflies_direct_sun (smooth) | 653.88 | 13.66 | 8.66e-04 | Smooth | Main + Interaction | Low |
| M7 | Single: temp_at_max_count_t_1 (smooth) | 653.91 | 13.68 | 8.54e-04 | Smooth | Single predictor | Low |
| M52 | Additive + Interaction: wind_max_gust + sum_butterflies_direct_sun + wind_max_gust×sum_butterflies_direct_sun (linear) | 654.45 | 14.23 | 6.51e-04 | Linear | Main + Interaction | Low |
| M41 | Additive + Interaction: temp_max + temp_at_max_count_t_1 + temp_max×temp_at_max_count_t_1 (smooth) | 654.72 | 14.50 | 5.67e-04 | Smooth | Main + Interaction | Low |
| M3 | Single: temp_min (smooth) | 655.44 | 15.21 | 3.97e-04 | Smooth | Single predictor | Low |
| M9 | Single: wind_max_gust (smooth) | 655.52 | 15.30 | 3.81e-04 | Smooth | Single predictor | Low |
| M53 | Additive: All temperature (smooth) | 655.96 | 15.73 | 3.06e-04 | Smooth | Additive | Low |
| M67 | All temp + all temp interactions (smooth) | 656.75 | 16.53 | 2.06e-04 | Smooth | Complex | Moderate |
| M20 | Interaction: temp_min × sum_butterflies_direct_sun (linear) | 656.77 | 16.55 | 2.04e-04 | Linear | Interaction only | Low |
| M35 | Additive + Interaction: temp_min + temp_at_max_count_t_1 + temp_min×temp_at_max_count_t_1 (smooth) | 657.01 | 16.79 | 1.81e-04 | Smooth | Main + Interaction | Low |
| M61 | Additive: temp_at_max_count_t_1 + wind_max_gust (smooth) | 657.30 | 17.08 | 1.56e-04 | Smooth | Additive | Low |
| M57 | Additive: temp_min + wind_max_gust (smooth) | 657.68 | 17.46 | 1.29e-04 | Smooth | Additive | Low |
| M46 | Additive + Interaction: temp_max + sum_butterflies_direct_sun + temp_max×sum_butterflies_direct_sun (linear) | 657.70 | 17.47 | 1.28e-04 | Linear | Main + Interaction | Low |
| M30 | Interaction: temp_at_max_count_t_1 × sum_butterflies_direct_sun (linear) | 657.90 | 17.68 | 1.16e-04 | Linear | Interaction only | Low |
| M44 | Additive + Interaction: temp_max + wind_max_gust + temp_max×wind_max_gust (linear) | 658.01 | 17.79 | 1.10e-04 | Linear | Main + Interaction | Low |
| M6 | Single: temp_max (linear) | 658.09 | 17.87 | 1.05e-04 | Linear | Single predictor | Low |
| M47 | Additive + Interaction: temp_at_max_count_t_1 + wind_max_gust + temp_at_max_count_t_1×wind_max_gust (smooth) | 658.22 | 17.99 | 9.90e-05 | Smooth | Main + Interaction | Low |
| M24 | Interaction: temp_max × wind_max_gust (linear) | 658.24 | 18.02 | 9.77e-05 | Linear | Interaction only | Low |
| M40 | Additive + Interaction: temp_min + sum_butterflies_direct_sun + temp_min×sum_butterflies_direct_sun (linear) | 658.42 | 18.20 | 8.92e-05 | Linear | Main + Interaction | Low |
| M37 | Additive + Interaction: temp_min + wind_max_gust + temp_min×wind_max_gust (smooth) | 658.61 | 18.39 | 8.12e-05 | Smooth | Main + Interaction | Low |
| M34 | Additive + Interaction: temp_min + temp_max + temp_min×temp_max (linear) | 658.93 | 18.71 | 6.91e-05 | Linear | Main + Interaction | Low |
| M65 | Additive: All predictors (additive) (smooth) | 659.21 | 18.98 | 6.03e-05 | Smooth | Additive | Low |
| M63 | Additive: All temp + wind (smooth) | 659.30 | 19.07 | 5.76e-05 | Smooth | Additive | Low |
| M26 | Interaction: temp_max × sum_butterflies_direct_sun (linear) | 659.93 | 19.71 | 4.21e-05 | Linear | Interaction only | Low |
| M56 | Additive: temp_min + temp_max (linear) | 660.55 | 20.32 | 3.08e-05 | Linear | Additive | Low |
| M60 | Additive: temp_max + wind_max_gust (linear) | 660.76 | 20.54 | 2.77e-05 | Linear | Additive | Low |
| M16 | Interaction: temp_min × temp_at_max_count_t_1 (linear) | 660.78 | 20.55 | 2.75e-05 | Linear | Interaction only | Low |
| M18 | Interaction: temp_min × wind_max_gust (linear) | 661.01 | 20.79 | 2.44e-05 | Linear | Interaction only | Low |
| M14 | Interaction: temp_min × temp_max (linear) | 661.32 | 21.10 | 2.10e-05 | Linear | Interaction only | Low |
| M28 | Interaction: temp_at_max_count_t_1 × wind_max_gust (linear) | 661.49 | 21.27 | 1.92e-05 | Linear | Interaction only | Low |
| M2 | Null (linear baseline) | 661.63 | 21.41 | 1.79e-05 | Linear | Null | Low |
| M22 | Interaction: temp_max × temp_at_max_count_t_1 (linear) | 662.25 | 22.03 | 1.31e-05 | Linear | Interaction only | Low |
| M12 | Single: sum_butterflies_direct_sun (linear) | 662.35 | 22.13 | 1.25e-05 | Linear | Single predictor | Low |
| M42 | Additive + Interaction: temp_max + temp_at_max_count_t_1 + temp_max×temp_at_max_count_t_1 (linear) | 662.96 | 22.74 | 9.24e-06 | Linear | Main + Interaction | Low |
| M4 | Single: temp_min (linear) | 663.33 | 23.10 | 7.69e-06 | Linear | Single predictor | Low |
| M50 | Additive + Interaction: temp_at_max_count_t_1 + sum_butterflies_direct_sun + temp_at_max_count_t_1×sum_butterflies_direct_sun (linear) | 663.53 | 23.31 | 6.93e-06 | Linear | Main + Interaction | Low |
| M54 | Additive: All temperature (linear) | 663.59 | 23.36 | 6.75e-06 | Linear | Additive | Low |
| M10 | Single: wind_max_gust (linear) | 663.79 | 23.57 | 6.09e-06 | Linear | Single predictor | Low |
| M69 | All additive + all temp×wind interactions (smooth) | 663.97 | 23.74 | 5.58e-06 | Smooth | Complex | Moderate |
| M8 | Single: temp_at_max_count_t_1 (linear) | 664.37 | 24.15 | 4.56e-06 | Linear | Single predictor | Low |
| M68 | All temp + all temp interactions (linear) | 664.52 | 24.30 | 4.23e-06 | Linear | Complex | Moderate |
| M58 | Additive: temp_min + wind_max_gust (linear) | 665.47 | 25.25 | 2.63e-06 | Linear | Additive | Low |
| M38 | Additive + Interaction: temp_min + wind_max_gust + temp_min×wind_max_gust (linear) | 666.02 | 25.80 | 1.99e-06 | Linear | Main + Interaction | Low |
| M36 | Additive + Interaction: temp_min + temp_at_max_count_t_1 + temp_min×temp_at_max_count_t_1 (linear) | 666.63 | 26.41 | 1.48e-06 | Linear | Main + Interaction | Low |
| M62 | Additive: temp_at_max_count_t_1 + wind_max_gust (linear) | 666.68 | 26.46 | 1.43e-06 | Linear | Additive | Low |
| M64 | Additive: All temp + wind (linear) | 666.83 | 26.61 | 1.33e-06 | Linear | Additive | Low |
| M48 | Additive + Interaction: temp_at_max_count_t_1 + wind_max_gust + temp_at_max_count_t_1×wind_max_gust (linear) | 666.96 | 26.73 | 1.25e-06 | Linear | Main + Interaction | Low |
| M66 | Additive: All predictors (additive) (linear) | 668.00 | 27.78 | 7.41e-07 | Linear | Additive | Low |
| M70 | All additive + all temp×wind interactions (linear) | 670.90 | 30.68 | 1.74e-07 | Linear | Complex | Moderate |
| M72 | FULL MODEL: All terms + all interactions (linear) | 710.96 | 70.74 | 3.49e-16 | Linear | Full model | High |
| M71 | FULL MODEL: All terms + all interactions (smooth) | 712.61 | 72.39 | 1.53e-16 | Smooth | Full model | High |
9.2 24-Hour Window: All 76 Models
| model | description | AICc | delta_AICc | weight_AICc | baseline_type | model_category | overfitting_risk |
|---|---|---|---|---|---|---|---|
| M31 | Interaction: wind_max_gust × sum_butterflies_direct_sun (smooth) | 636.33 | 0.00 | 5.07e-01 | Smooth | Interaction only | Low |
| M23 | Interaction: temp_max × wind_max_gust (smooth) | 639.92 | 3.59 | 8.44e-02 | Smooth | Interaction only | Low |
| M29 | Interaction: temp_at_max_count_t_1 × sum_butterflies_direct_sun (smooth) | 640.76 | 4.42 | 5.55e-02 | Smooth | Interaction only | Low |
| M19 | Interaction: temp_min × sum_butterflies_direct_sun (smooth) | 641.11 | 4.78 | 4.66e-02 | Smooth | Interaction only | Low |
| M51 | Additive + Interaction: wind_max_gust + sum_butterflies_direct_sun + wind_max_gust×sum_butterflies_direct_sun (smooth) | 641.72 | 5.39 | 3.42e-02 | Smooth | Main + Interaction | Low |
| M45 | Additive + Interaction: temp_max + sum_butterflies_direct_sun + temp_max×sum_butterflies_direct_sun (smooth) | 641.80 | 5.46 | 3.30e-02 | Smooth | Main + Interaction | Low |
| M5 | Single: temp_max (smooth) | 642.19 | 5.86 | 2.71e-02 | Smooth | Single predictor | Low |
| M67 | All temp + all temp interactions (smooth) | 642.49 | 6.16 | 2.33e-02 | Smooth | Complex | Moderate |
| M43 | Additive + Interaction: temp_max + wind_max_gust + temp_max×wind_max_gust (smooth) | 642.69 | 6.36 | 2.11e-02 | Smooth | Main + Interaction | Low |
| M17 | Interaction: temp_min × wind_max_gust (smooth) | 642.73 | 6.40 | 2.07e-02 | Smooth | Interaction only | Low |
| M25 | Interaction: temp_max × sum_butterflies_direct_sun (smooth) | 642.75 | 6.42 | 2.05e-02 | Smooth | Interaction only | Low |
| M33 | Additive + Interaction: temp_min + temp_max + temp_min×temp_max (smooth) | 642.90 | 6.57 | 1.90e-02 | Smooth | Main + Interaction | Low |
| M41 | Additive + Interaction: temp_max + temp_at_max_count_t_1 + temp_max×temp_at_max_count_t_1 (smooth) | 643.73 | 7.40 | 1.26e-02 | Smooth | Main + Interaction | Low |
| M15 | Interaction: temp_min × temp_at_max_count_t_1 (smooth) | 643.79 | 7.46 | 1.22e-02 | Smooth | Interaction only | Low |
| M32 | Interaction: wind_max_gust × sum_butterflies_direct_sun (linear) | 644.33 | 8.00 | 9.30e-03 | Linear | Interaction only | Low |
| M53 | Additive: All temperature (smooth) | 644.61 | 8.28 | 8.09e-03 | Smooth | Additive | Low |
| M27 | Interaction: temp_at_max_count_t_1 × wind_max_gust (smooth) | 644.95 | 8.62 | 6.82e-03 | Smooth | Interaction only | Low |
| M59 | Additive: temp_max + wind_max_gust (smooth) | 645.02 | 8.69 | 6.59e-03 | Smooth | Additive | Low |
| M55 | Additive: temp_min + temp_max (smooth) | 645.02 | 8.69 | 6.58e-03 | Smooth | Additive | Low |
| M21 | Interaction: temp_max × temp_at_max_count_t_1 (smooth) | 645.52 | 9.19 | 5.13e-03 | Smooth | Interaction only | Low |
| M1 | Null (smooth baseline) | 645.54 | 9.21 | 5.07e-03 | Smooth | Null | Low |
| M13 | Interaction: temp_min × temp_max (smooth) | 645.90 | 9.57 | 4.23e-03 | Smooth | Interaction only | Low |
| M39 | Additive + Interaction: temp_min + sum_butterflies_direct_sun + temp_min×sum_butterflies_direct_sun (smooth) | 646.00 | 9.66 | 4.04e-03 | Smooth | Main + Interaction | Low |
| M7 | Single: temp_at_max_count_t_1 (smooth) | 646.19 | 9.86 | 3.67e-03 | Smooth | Single predictor | Low |
| M49 | Additive + Interaction: temp_at_max_count_t_1 + sum_butterflies_direct_sun + temp_at_max_count_t_1×sum_butterflies_direct_sun (smooth) | 646.79 | 10.46 | 2.72e-03 | Smooth | Main + Interaction | Low |
| M3 | Single: temp_min (smooth) | 647.30 | 10.97 | 2.10e-03 | Smooth | Single predictor | Low |
| M24 | Interaction: temp_max × wind_max_gust (linear) | 647.38 | 11.05 | 2.02e-03 | Linear | Interaction only | Low |
| M9 | Single: wind_max_gust (smooth) | 647.82 | 11.49 | 1.62e-03 | Smooth | Single predictor | Low |
| M11 | Single: sum_butterflies_direct_sun (smooth) | 648.02 | 11.69 | 1.47e-03 | Smooth | Single predictor | Low |
| M30 | Interaction: temp_at_max_count_t_1 × sum_butterflies_direct_sun (linear) | 648.22 | 11.89 | 1.33e-03 | Linear | Interaction only | Low |
| M35 | Additive + Interaction: temp_min + temp_at_max_count_t_1 + temp_min×temp_at_max_count_t_1 (smooth) | 648.40 | 12.06 | 1.22e-03 | Smooth | Main + Interaction | Low |
| M63 | Additive: All temp + wind (smooth) | 648.58 | 12.25 | 1.11e-03 | Smooth | Additive | Low |
| M20 | Interaction: temp_min × sum_butterflies_direct_sun (linear) | 648.79 | 12.46 | 9.98e-04 | Linear | Interaction only | Low |
| M68 | All temp + all temp interactions (linear) | 648.97 | 12.64 | 9.13e-04 | Linear | Complex | Moderate |
| M37 | Additive + Interaction: temp_min + wind_max_gust + temp_min×wind_max_gust (smooth) | 649.34 | 13.01 | 7.58e-04 | Smooth | Main + Interaction | Low |
| M61 | Additive: temp_at_max_count_t_1 + wind_max_gust (smooth) | 649.36 | 13.03 | 7.51e-04 | Smooth | Additive | Low |
| M47 | Additive + Interaction: temp_at_max_count_t_1 + wind_max_gust + temp_at_max_count_t_1×wind_max_gust (smooth) | 649.59 | 13.26 | 6.70e-04 | Smooth | Main + Interaction | Low |
| M52 | Additive + Interaction: wind_max_gust + sum_butterflies_direct_sun + wind_max_gust×sum_butterflies_direct_sun (linear) | 649.81 | 13.48 | 6.01e-04 | Linear | Main + Interaction | Low |
| M18 | Interaction: temp_min × wind_max_gust (linear) | 650.19 | 13.86 | 4.96e-04 | Linear | Interaction only | Low |
| M57 | Additive: temp_min + wind_max_gust (smooth) | 650.20 | 13.86 | 4.95e-04 | Smooth | Additive | Low |
| M26 | Interaction: temp_max × sum_butterflies_direct_sun (linear) | 650.34 | 14.01 | 4.61e-04 | Linear | Interaction only | Low |
| M44 | Additive + Interaction: temp_max + wind_max_gust + temp_max×wind_max_gust (linear) | 650.36 | 14.03 | 4.56e-04 | Linear | Main + Interaction | Low |
| M16 | Interaction: temp_min × temp_at_max_count_t_1 (linear) | 650.75 | 14.42 | 3.75e-04 | Linear | Interaction only | Low |
| M6 | Single: temp_max (linear) | 651.09 | 14.76 | 3.16e-04 | Linear | Single predictor | Low |
| M46 | Additive + Interaction: temp_max + sum_butterflies_direct_sun + temp_max×sum_butterflies_direct_sun (linear) | 651.41 | 15.08 | 2.70e-04 | Linear | Main + Interaction | Low |
| M42 | Additive + Interaction: temp_max + temp_at_max_count_t_1 + temp_max×temp_at_max_count_t_1 (linear) | 652.00 | 15.67 | 2.01e-04 | Linear | Main + Interaction | Low |
| M65 | Additive: All predictors (additive) (smooth) | 652.07 | 15.73 | 1.94e-04 | Smooth | Additive | Low |
| M28 | Interaction: temp_at_max_count_t_1 × wind_max_gust (linear) | 652.41 | 16.08 | 1.64e-04 | Linear | Interaction only | Low |
| M34 | Additive + Interaction: temp_min + temp_max + temp_min×temp_max (linear) | 652.81 | 16.48 | 1.34e-04 | Linear | Main + Interaction | Low |
| M22 | Interaction: temp_max × temp_at_max_count_t_1 (linear) | 652.98 | 16.65 | 1.23e-04 | Linear | Interaction only | Low |
| M2 | Null (linear baseline) | 653.04 | 16.71 | 1.20e-04 | Linear | Null | Low |
| M54 | Additive: All temperature (linear) | 653.07 | 16.74 | 1.17e-04 | Linear | Additive | Low |
| M14 | Interaction: temp_min × temp_max (linear) | 653.40 | 17.06 | 1.00e-04 | Linear | Interaction only | Low |
| M56 | Additive: temp_min + temp_max (linear) | 653.55 | 17.21 | 9.27e-05 | Linear | Additive | Low |
| M60 | Additive: temp_max + wind_max_gust (linear) | 653.72 | 17.39 | 8.51e-05 | Linear | Additive | Low |
| M40 | Additive + Interaction: temp_min + sum_butterflies_direct_sun + temp_min×sum_butterflies_direct_sun (linear) | 653.82 | 17.49 | 8.08e-05 | Linear | Main + Interaction | Low |
| M69 | All additive + all temp×wind interactions (smooth) | 654.00 | 17.67 | 7.39e-05 | Smooth | Complex | Moderate |
| M50 | Additive + Interaction: temp_at_max_count_t_1 + sum_butterflies_direct_sun + temp_at_max_count_t_1×sum_butterflies_direct_sun (linear) | 654.12 | 17.79 | 6.96e-05 | Linear | Main + Interaction | Low |
| M8 | Single: temp_at_max_count_t_1 (linear) | 654.78 | 18.45 | 5.01e-05 | Linear | Single predictor | Low |
| M4 | Single: temp_min (linear) | 654.82 | 18.49 | 4.90e-05 | Linear | Single predictor | Low |
| M10 | Single: wind_max_gust (linear) | 655.34 | 19.01 | 3.78e-05 | Linear | Single predictor | Low |
| M12 | Single: sum_butterflies_direct_sun (linear) | 655.95 | 19.62 | 2.78e-05 | Linear | Single predictor | Low |
| M36 | Additive + Interaction: temp_min + temp_at_max_count_t_1 + temp_min×temp_at_max_count_t_1 (linear) | 656.03 | 19.70 | 2.68e-05 | Linear | Main + Interaction | Low |
| M64 | Additive: All temp + wind (linear) | 656.48 | 20.15 | 2.14e-05 | Linear | Additive | Low |
| M38 | Additive + Interaction: temp_min + wind_max_gust + temp_min×wind_max_gust (linear) | 656.56 | 20.22 | 2.06e-05 | Linear | Main + Interaction | Low |
| M62 | Additive: temp_at_max_count_t_1 + wind_max_gust (linear) | 657.31 | 20.98 | 1.41e-05 | Linear | Additive | Low |
| M48 | Additive + Interaction: temp_at_max_count_t_1 + wind_max_gust + temp_at_max_count_t_1×wind_max_gust (linear) | 657.32 | 20.99 | 1.40e-05 | Linear | Main + Interaction | Low |
| M58 | Additive: temp_min + wind_max_gust (linear) | 657.60 | 21.27 | 1.22e-05 | Linear | Additive | Low |
| M66 | Additive: All predictors (additive) (linear) | 659.86 | 23.53 | 3.94e-06 | Linear | Additive | Low |
| M70 | All additive + all temp×wind interactions (linear) | 660.18 | 23.85 | 3.36e-06 | Linear | Complex | Moderate |
| M71 | FULL MODEL: All terms + all interactions (smooth) | 699.80 | 63.47 | 8.37e-15 | Smooth | Full model | High |
| M72 | FULL MODEL: All terms + all interactions (linear) | 703.91 | 67.58 | 1.07e-15 | Linear | Full model | High |
10 References
- Original analysis: Daily GAM Results
- Sunset window detailed analysis: sunset_window_gam_analysis.html
- 24-hour window detailed analysis: 24hr_window_gam_analysis.html
- Data sources:
data/monarch_daily_lag_analysis_sunset_window.csvdata/monarch_daily_lag_analysis_24hr_window.csv
Document generated: 2025-10-03